A quadratic equation is given by \( x^2 - 5x + 6 = 0 \). Find the values of \( x \).

["# Solving the Quadratic Equation ( x^2 - 5x + 6 = 0 ): Step-by-Step Guide", "A quadratic equation is a fundamental concept in algebra, widely used in mathematics, physics, engineering, and many other fields. Understanding how to solve these equations is essential for anyone studying mathematics or related disciplines. In this article, we will explore how to find the values of ( x ) in the quadratic equation:", "[\nx^2 - 5x + 6 = 0\n]", "We’ll walk through the process using multiple methods—factoring, the quadratic formula, and completing the square—so you can choose the approach that best fits your learning style.", "---", "## What is a Quadratic Equation?", "A quadratic equation is any equation of the form:", "[\nax^2 + bx + c = 0\n]", "where ( a ), ( b ), and ( c ) are real numbers and ( a <br/>\neq 0 ). The general solutions to quadratic equations yield either two real roots, one repeated real root, or two complex numbers, depending on the discriminant.", "---", "## Method 1: Factoring", "Factoring is often the quickest way to solve quadratic equations when the expression can be easily broken into binomials.", "### Step 1: Identify coefficients", "For the equation:", "[\nx^2 - 5x + 6 = 0\n]", "We see that:\n- ( a = 1 )\n- ( b = -5 )\n- ( c = 6 )", "### Step 2: Find two numbers that multiply to ( c = 6 ) and add to ( b = -5 )", "We look for two numbers whose product is ( 6 ) and sum is ( -5 ). These numbers are ( -2 ) and ( -3 ), since:", "[\n-2 \ imes -3 = 6 \quad \ ext{and} \quad -2 + (-3) = -5\n]", "### Step 3: Rewrite the equation in factored form", "Using these numbers, we factor the quadratic:", "[\nx^2 - 5x + 6 = (x - 2)(x - 3) = 0\n]", "### Step 4: Apply the zero product property", "If a product equals zero, then one of the factors must be zero:", "[\nx - 2 = 0 \quad \ ext{or} \quad x - 3 = 0\n]", "Solving each:", "- ( x = 2 )\n- ( x = 3 )", "---", "## Method 2: Quadratic Formula", "When factoring is difficult or impossible, the quadratic formula provides a reliable method for finding solutions:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "### Step 1: Identify coefficients", "Again, we have:", "- ( a = 1 ), ( b = -5 ), ( c = 6 )", "### Step 2: Compute the discriminant", "The discriminant ( D ) is:", "[\nD = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1\n]", "Since ( D = 1 > 0 ), there are two distinct real roots.", "### Step 3: Plug values into the formula", "[\nx = \frac{-(-5) \pm \sqrt{1}}{2(1)} = \frac{5 \pm 1}{2}\n]", "So:", "- ( x = \frac{5 + 1}{2} = \frac{6}{2} = 3 )\n- ( x = \frac{5 - 1}{2} = \frac{4}{2} = 2 )", "---", "## Method 3: Completing the Square", "This algebraic method transforms the equation into a perfect square trinomial, making solutions easier to extract.", "### Step 1: Start with the equation", "[\nx^2 - 5x + 6 = 0\n]", "Move the constant term to the right:", "[\nx^2 - 5x = -6\n]", "### Step 2: Complete the square", "Take half of the coefficient of ( x ), which is ( -5 ), so half is ( -\frac{5}{2} ), and square it:", "[\n\left(-\frac{5}{2}\right)^2 = \frac{25}{4}\n]", "Add this to both sides:", "[\nx^2 - 5x + \frac{25}{4} = -6 + \frac{25}{4}\n]", "Simplify the right-hand side:", "[\n-6 + \frac{25}{4} = -\frac{24}{4} + \frac{25}{4} = \frac{1}{4}\n]", "So:", "[\n\left(x - \frac{5}{2}\right)^2 = \frac{1}{4}\n]", "### Step 3: Solve by taking square roots", "[\nx - \frac{5}{2} = \pm \sqrt{\frac{1}{4}} = \pm \frac{1}{2}\n]", "Thus:", "- ( x = \frac{5}{2} + \frac{1}{2} = 3 )\n- ( x = \frac{5}{2} - \frac{1}{2} = 2 )", "---", "## Summary of Solutions", "All three methods consistently yield the same solutions:", "[\n\boxed{x = 2} \quad \ ext{and} \quad \boxed{x = 3}\n]", "These are the values of ( x ) that satisfy the quadratic equation ( x^2 - 5x + 6 = 0 ).", "---", "## Why Is Solving Quadratics Important?", "Understanding how to solve ( ax^2 + bx + c = 0 ) is crucial because quadratic equations model numerous real-world phenomena, including projectile motion, area problems (quadratic areas), and optimization scenarios in economics and engineering. Mastering these techniques strengthens algebraic thinking and problem-solving skills.", "---", "## Further Practice", "Try solving other similar quadratic equations:", "- ( x^2 + 4x + 3 = 0 )\n- ( 2x^2 - 8x + 6 = 0 )\n- ( x^2 - 3x - 4 = 0 )", "Use any of the three methods—factoring, quadratic formula, or completing the square—to reinforce your understanding.", "---", "## Final Thoughts", "Whether you prefer factoring for speed, the quadratic formula for certainty, or completing the square for deeper insight, mastering these methods allows you to confidently solve any quadratic equation. The equation ( x^2 - 5x + 6 = 0 ) serves as a perfect example to practice and internalize these powerful algebraic tools.", "Start with practicing today—your math skills will grow stronger with every problem solved!"]